Blum axioms

In computational complexity theory the Blum axioms or Blum complexity axioms are axioms that specify desirable properties of complexity measures on the set of computable functions. The axioms were first defined by Manuel Blum in 1967.

Importantly, Blum's speedup theorem and the Gap theorem hold for any complexity measure satisfying these axioms. The most well-known measures satisfying these axioms are those of time (i.e., running time) and space (i.e., memory usage).

A Blum complexity measure is a pair ${\displaystyle (\varphi ,\Phi )}$ with ${\displaystyle \varphi }$ a Gödel numbering of the partial computable functions ${\displaystyle \mathbf {P} ^{(1)}}$ and a computable function

which satisfies the following Blum axioms. We write ${\displaystyle \varphi _{i}}$ for the i-th partial computable function under the Gödel numbering ${\displaystyle \varphi }$, and ${\displaystyle \Phi _{i}}$ for the partial computable function ${\displaystyle \Phi (i)}$.

A Blum complexity measure is defined using computable functions without any reference to a specific model of computation. In order to make the definition more accessible we rephrase the Blum axioms in terms of Turing machines:

A Blum complexity measure is a function ${\displaystyle \Phi }$ from pairs (Turing machine ${\displaystyle M}$, input ${\displaystyle x}$ for ${\displaystyle M}$) to the natural numbers union infinity. Furthermore, ${\displaystyle \Phi }$ should satisfy the following axioms:

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